Change of basis

A change of basis is a method to convert vector coordinates with respect to one basis to coordinates with respect to another basis.

Consider a vector u with reference to orthogonal unit basis vectors x1, x2 and x3.

\textbf{\textit{u}}=px_1+qx_2+rx_3\; \; \; \; \; \; \; (1)

where p, q and r are the scalar components of the unit basis vectors x1, x2 and x3 respectively. Clearly,

p=\textbf{\textit{u}}\cdot x_1\; \; \; \; q=\textbf{\textit{u}}\cdot x_2\; \; \; \; r=\textbf{\textit{u}}\cdot x_3\; \; \; \; (2)

If we define u with respect to another orthogonal reference frame with unit basis vectors x’1, x’2 and x’3 (see above diagram), we have:

\textbf{\textit{u}}=p' x'_1+q'x'_2+r'x'_3\; \; \; \; (3)


p'=\textbf{\textit{u}}\cdot x'_1\; \; \; \; q'=\textbf{\textit{u}}\cdot x'_2\; \; \; \; r'=\textbf{\textit{u}}\cdot x'_3\; \; \; \; (4)

To find the relationship between the two reference frames, we substitute eq1 in eq4 to give:

p'=\left ( px_1+qx_2+rx_3 \right )\cdot x'_1=px_1\cdot x'_1+qx_2\cdot x'_1+rx_3\cdot x'_1\; \; \; \; \; \; \; (5)

q'=\left ( px_1+qx_2+rx_3 \right )\cdot x'_2=px_1\cdot x'_2+qx_2\cdot x'_2+rx_3\cdot x'_2\; \; \; \; \; \; \; (6)

r'=\left ( px_1+qx_2+rx_3 \right )\cdot x'_3=px_1\cdot x'_3+qx_2\cdot x'_3+rx_3\cdot x'_3\; \; \; \; \; \; \; (7)

Since the dot product of two vectors gives a scalar, we can write eq5, eq6 and eq7 as:




which can be expressed in the following matrix equation:

\begin{pmatrix} p'\\ q'\\ r' \end{pmatrix}=\begin{pmatrix} a_{11} &a_{12} &a_{13} \\ a_{21} & a_{22} &a_{23} \\ a_{31} &a_{32} & a_{33} \end{pmatrix}\begin{pmatrix} p\\ q\\ r \end{pmatrix}\; \; \; \; \; \; \; (8)

where a_{ij}=x'_i\cdot x_j=\left | x'_i \right |\left | x_j \right |cos\theta_{x'_i,x_j}=cos\theta _{x'_i,x_j}. The matrix containing ij is known as the change of basis matrix . We can rewrite eq8 as:

u'_i=\sum_{j=1,2,3}a_{ij}u_j\; \; \; \; \; \; \; (9)

where u'_1=p',u'_2=q',u'_3=r' and u_1=p,u_2=q,u_3=r. Eq9 is often written as a short form by omitting the summation symbol:

u'_i=a_{ij}u_j\; \; \; \; \; \; \; (10)

For example, the change of basis from the orthogonal basis xj to the orthogonal basis  x’i as a rotation about the x3-axis, where x3 coincides with x’3 (x’3 = x3 and is perpendicular to the plane of the page), is depicted as:

To determine the change of basis matrix, we have

x'_1\cdot x_3=x'_3\cdot x_1=x'_2\cdot x_3=x'_3\cdot x_2=cos90^o=0

x'_3\cdot x_3=cos0^o=1

x'_1\cdot x_1=x'_2\cdot x_2=cos\theta

x'_1\cdot x_2=cos\left ( 90^o-\theta \right )=sin\theta

x'_2\cdot x_1=cos\left ( 90^o+\theta \right )=-sin\theta


a_{ij}=\begin{pmatrix} a_{11} & a_{12}&a_{13}\\ a_{21}& a_{22}&a_{23}\\ a_{31}& a_{32}& a_{33}\end{pmatrix}=\begin{pmatrix} x'_1\cdot x_1 & x'_1\cdot x_2&x'_1\cdot x_3\\ x'_2\cdot x_1&x'_2\cdot x_2 &x'_2\cdot x_3\\ x'_3\cdot x_1& x'_3\cdot x_2& x'_3\cdot x_3\end{pmatrix}=\begin{pmatrix} cos\theta &sin\theta &0 \\ -sin\theta & cos\theta &0 \\ 0 &0 &1 \end{pmatrix}\; \; \; \; \; \; \; (12)

Consider again the vector u with reference to a coordinate system that is defined by the orthogonal unit basis vectors x1, x2 and x3  (see diagram below).

The rotation of u with respect to this fixed coordinate system is expressed as:

Compared to the change of basis of u that was described earlier, the rotation matrix R is the inverse of the change of basis matrix (similarly, a reflection matrix is the inverse of the change of basis matrix by a reflection). In other words, the rotation of u by θ can be perceived a change of basis, with the coordinate system rotated by –θ. This implies that we can analyse a problem in two ways.


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